Numerical Models for Field Problems

CFU: 9

Prerequisites

None.

Preliminary Courses

None.

Learning Goals

The course aims at illustrating the fundamental aspects of numerical modeling of interest to an electrical and information engineer, providing the basic tools for solving field problems with the computer. The approach aims to mediate between a rigorous mathematical approach and the need to lead students to solve practical problems directly related to their specific background and interests. The MATLAB® programming language is used in the numerical laboratory.

Expected Learning Outcomes

Knowledge and understanding

The course aims to provide students with the knowledge and methodological tools necessary to tackle the resolution of a field problem on the computer and critically evaluate the expected characteristics of a numerical solution of a field problem, such as that which can also be obtained directly by means of a commercial code.

Applying knowledge and understanding

The student must demonstrate to be able to concretely use the acquired knowledge, proving to be able to apply it to the solution of a field problem on the computer and to the critical evaluation of the expected characteristics of the numerical solution, including the one obtained with commercial codes.

Course Content - Syllabus

Recalls of the fundamental concepts and results from Linear Algebra

Linear spaces of finite dimensions. Matrices. Symmetric, Hermitian, normal, unitary, orthogonal matrices. Determinant. Eigenvalues and eigenvectors. Linearly independent eigenvectors. Diagonalization. Localization of the eigenvalues. Gershgorin's first and second theorem. Diagonal predominance. Positive definite matrices. Scalar product. Vector norms. Rayleigh – Ritz quotient. Equivalence of norms. Continuity of the norm. Matrix norms. Decomposition to singular values (SVD). Conditioning of a matrix.

Differential and integral problems

General information on the models described by partial differential equations. Characteristic lines. Classification of quasi-linear equations. Characteristic lines for hyperbolic equations. Introduction to integral equations. Integral formulation of the external problem for the electrostatic potential. Application example: the problem of electrostatic resonances of an object of uniform permittivity.

Finite Difference Method

Approximation of the first and second derivative. Solution of the Poisson equation with the finite difference method. Consistency, stability and convergence.

Finite Elements Method

Formulations of the field problem: strong and weak form; variational formulations. Introduction to the Finite Element Method. Poisson equation. Polynomial interpolation. Lagrange polynomials. Piece-wise linear splines. Interpolation error. Variational formulations and weak formulations. Galerkin's method. Convergence of the Finite Element Method. Linear shape functions and barycentric coordinates. Iso-parametric elements of order greater than 1.

Numerical integration

Numerical integration. Rectangle formula; trapezoid formula, Simpson's formula. Discretization error. Gauss Legendre formulas.

Systems of linear algebraic equations

Solving systems of algebraic equations. Direct methods. The method of elimination of Gauss with partial Pivot. LU factorization. Factorization by means of a succession of matrices. LL^H factorization. Cholesky's Method. Sparse matrices and banded matrices. Introduction to the problem of bandwidth reduction. Introduction to the the problem of conditioning and numerical stability. Solution of systems of least squares algebraic equations. Normal equations. Numerical solutions of normal equations. Solution using the QR method. Solution by decomposition into singular values. Pseudo-inverse matrix. Introduction to the solution of constrained optimization problems with the method of Lagrange multipliers. Outline of Tihonov's regularization. Solving linear systems with iterative methods. Convergence of the iterative method. Speed of convergence. Criteria for stopping the iteration. Jacobi and Gauss-Seidel methods. The relaxation method. Convergence and error estimation. The gradient and conjugate gradient method.

Systems of Non-linear Algebraic Equations

Systems of non-linear algebraic equations. Bisection method. Fixed point iteration. Newton Raphson's method. Convergence, error estimate, convergence speed.

Systems of first order differential equations with ordinary derivatives

Numerical methods. Series expansion methods. Euler's method. Local discretization error. Consistency of the method. Convergence study. Global error and numerical stability. The implicit Euler method. The theta method. Influence of round off errors. Outline on the methods of Runge-Kutta.

Hints on Machine Learning

Introduction to machine learning. Elements of unsupervised learning: principal component analysis. Neural Networks: definitions, topology. Perceptron with one or more layers. Universal interpolation property. Stochastic Gradient Descent (SGD) algorithm. Introduction to MATLAB® toolbox for designing machine learning algorithms based on neural networks. Numerical solution of interpolation problems with neural networks. Back Propagation (BP) algorithm for calculating the gradient of the loss function of a neural network. Automatic Differentiation (AD): forward mode and backward mode. Computational graph. Connections between AD and BP.

On the numerical solution of Maxwell equations

Maxwell's equations in the quasi-stationary limit. The diffusion equation of the magnetic field. Solution with the finite difference method. Euler's method explicit, implicit and theta. Stability. Wave equations. D'Alembert's formula. Explicit integration. Stability analysis. Courant-Friedrichs-Lewy condition. The problem of numerical dispersion. Finite element formulations for Maxwell's equations. An outline of edge elements.

Readings/Bibliography

Reference texts

F. Trevisan, F. Villone, Modelli numerici per campi e circuiti, SGE Padova
G. Miano, Modelli Numerici per i Campi, Napoli, settembre 2009, lecture notes available in pdf format on the website www.elettrotecnica.unina.it on the course page of prof.G. Miano
Additional teaching material (including the software used in the laboratory hours) is available on the website www.elettrotecnica.unina.it and on TEAMS at the address of the course.Any further references may be given during the course.

Additional reference books

V. Comincioli. Analisi numerica: Metodi Modelli Applicazioni. Nuova edizione, in formato e-book, 981 pp. Apogeo, Feltrinelli Milano, 2005
A. Quarteroni, R. Sacco, F. Saleri, P. Gervasio, Matematica Numerica, 4a edizione Springer 2014. [QS]
A. Quarteroni, Modellistica Numerica per Problemi Differenziali, 6a edizione Springer 2016 [AQ]

Teaching Method

Lectures (about 85%), exercises and computer lab with use of software written in MATLAB® (about 15%).

Examination/Evaluation criteria

Exam type

Only oral and an optional project discussion.